Full text

(1)

Cihan Karabulut

The Graduate Center, City University of New York

This work is made publicly available by the City University of New York (CUNY).

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k,d

2,d

(10)

Chapter 1 Introduction

k,D

k,D

(x) := X

disc(Q)=D a<0<Q(x)

k−1

2

3

2

(11)

2,5

2,5

2,5

2,D

D

D

D

4,D

D

k,D

k,D

n≥1

f

2k−1

f

k,D

k,D

k,D

k,D

k,D

k,D

k,D

k,D

(12)

k,D

k,D

k,D

k,d

k,d

∆(h)=d a<0<h(z)

k−1

2

(13)

k,d

k,d

: = X

b21+b22−4ac=d a<0<c

k

= X

b21+b22<d b1+b2≡d(mod 2)

k

21

22

1

2

2,d

4,d

2,d

4,d

k,d

k,d

k,D

2,d

4,d

6,d

(14)

6,d

d

6,6

6,d

d

6,6

d

k,d

k,dj

j

j

k,d

k,d

3

3

K

3

(15)

K

3

K

cusp1

k,k

k

K

k,k

k

C

k

k

k

k,D

k,D

k,d

k,d

k,d

k,d

k,d

(16)

k,D

k,D

k,d

k,d

k,D

k,d

k,d

2,d

2,d

(17)

Zagier’s Function

k,D

(x) := X

disc(Q)=D a<0<Q(x)

k−1

2

3

2

2,D

4,D

D

D

D

D

D

D

k,D

(18)

k,D

k,D

k,D

2k−2

k,D

k,D

k,D

k,D

b2−4ac=D a>0>c

2

k−1

k,D

k,D

2,D

k,D

Q

a=1

k−1

k,D

(19)

2,5

2,5

1

2

1

2

1

2

0

1

2

0

1

j+1

j−1

j

j

2,5

2j

j

j+1

2j+1

j2

j

j+1

2j+1

j

2,5

j

(20)

j

j

1−j2

2,D

k,D

k,D

k

2−k

k

2−k

k−2

(21)

2−k

k

k+

k

2−k

k

k

k

1

2

1

2−k

2

2

1

2

k

k

k

2−k

k

k

2−k

k

k

1

k

k

2−k

1

k

1

k

k

(22)

cusp1

k

k

k

1

k

k

k

k

k,D

k

k

k

1

k

(23)

n

0

n

i∞

0

w

w

n=0

−n+1

n

w−n

k

+

0≤n≤w n even

n/2

n

w−n

0<n<w n odd

(n−1)/2

n

w−n

±

k±

+

2−k

2−k

2

k

k

k

2

k

2−k

2−k

2

k

k

k+

k

±

k

k±

(24)

k

k

+

k

k+

k−2

k

1

k

n=1

n

n

2πiz

k

k−1

n=1

n

k−1

n

k

k−1

k−1

2−k

k−1

k−1

k

k

−k

k−1

k−1

2−k

k−1

k−1

k

k−1

γ

f,γ

2−k

(25)

f,γ

k

f,γ

1

k

k

f,γ

g,γ

2−k

2−k

2−k

k

+

k−1

n=1

n

k−1

k−1

n=1

n

k−1

+

f,γ

+f,γ

+

2−k

f,γ

2−k

f,γ+

k+

k−2

f,γ

k

k,D

(26)

k,D

k,D

k

k−1/2

b2−4ac=D a,b,c∈Z

2

k

k

k,D

k,D

k,D

+

k,D

D

2k−2

2k−2

k,D

k,D

+

k,D

k+

k−2

k,D

k,D

D

n=0

k,D

2k−1

k,D

k,D

(27)

k,D

(28)

Sums of binary Hermitian forms

t

t

t

(29)

t

2

+

±

(30)

±

±

±

(31)

k,D

k,d

k,d

(z) : = X

∆(h)=d a<0<h(z,1)

k−1

|b|2−4ac=d a<0

k−1

k,d

k

(32)

k

0≤n,j≤k−1

n,j

n

j

n,j

k

1−k

k−1

k−1

2

2

2

−1

t

−1

t

(33)

k,d

k,d

1−k

k−1

k−1

k,d

= X

∆(h)=d a<0<γt(h)(z,1)

t

k−1

k,d

k,d

k,d

k,d

k,d

k,d

k,d

k,d

k,d

k,d

k,d

k,d

(−z) = X

∆(h)=d a<0<h(z,1)

k−1

|b|2−4ac=d a<0

k−1

|b|2−4ac=d a<0

k−1

2

k,d

k,d

k,d

k,d

(34)

k,d

k,d

k,d

k,d

1−k

k,d

1−k

T )(z) = X

∆(h)=d a<0<Tt(h)(z,1)

k−1

= X

∆(h)=d a<0<Tt(h)(z,1)

k−1

k,d

k,d

k,d

i

k,d

k,d

k,d

: = X

b21+b22−4ac=d a<0<c

k

= X

b21+b22<d b1+b2≡d(mod 2)

k

21

22

1

2

(35)

X

|b|< D b≡D(mod 2)

1

2

D

X

|b|< D b≡D(mod 2)

3

2

D

k,D

2,d

4,d

21

22

21

22

1

2

21

22

21

22

1

2

21

22

21

22

1

2

21

22

2,14

4,14

2,14

2,13

2,5

13

5

(36)

4,14

4,13

4,5

13

5

21

22

21

22

2,14

4,14

k,D

k,d

k,d

k,d

2,d

4,d

2,d

−1

2,d

2,d

2,d

2,d

2

2

2

1

2

2

2

2

2

2

2

1

2

2

2

(37)

1

2

2,d

2,d

−1

2,d

2,d

2,d

2,d

2,d

|b|2−4ac=d a<0

|b|2−4ac=d a<0

|b|2−4ac=d a<0

|b|2−4ac=d a<0

|b|2−4ac=d c<0

|b|2−4ac=d a<0

|b|2−4ac=d c<0<a

|b|2−4ac=d a<0<c

|b|2−4ac=d c<0<a

|b|2−4ac=d a<0<c

|b|2−4ac=d c<0<a

|b|2−4ac=d c<0<a

(38)

2,d

−1

2,d

|b|2−4ac=d c<0<a

2

2

2,d

2,d

2,d

−1

2,d

2,d

2,d

2,d

2,d0

2,d

2,d

2,d0

−1

2,d0

2,d0

−1

i

2,d0

2,d0

2,d0

−1

2,d

−1

2,d

2,d

2,d

2,d

2,d0

(39)

i

2,d0

2,d0

2,d

2,d

2,d

2,d

2,d

−1

k,d

k,d

1−k

|b|2−4ac=d c<0<a

k−1

k,d

k,d

k,d

k,d

k,d

k,d

1−k

k,d

k,d

2k−2

k,d

2

(40)

k,d

k,d

k,d

2

2

k,d

k,d

2k−2

k,d

k,d

2k−2

k,d

k,d

2k−2

k,d

k,d

2k−2

k,d

k,d

2k−2

k,d

2k−2

k,d

2k−2

k,d

4,d

−3

4,d

4,d

4,d

3

3

4,d

−3

4,d

4,d

|b|2−4ac=d c<0<a

3

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